research article european option pricing with transaction ...time points for three scenarios, that...

Research ArticleEuropean Option Pricing with Transaction Costs inLeacutevy Jump Environment

Jiayin Li1 Huisheng Shu1 and Xiu Kan2

1 School of Science Donghua University Shanghai 200051 China2 College of Electronic and Electrical Engineering Shanghai University of Engineering Science Shanghai 201620 China

Correspondence should be addressed to Huisheng Shu hsshudhueducn

Received 25 January 2014 Accepted 20 February 2014 Published 27 March 2014

Academic Editor Bo Shen

Copyright copy 2014 Jiayin Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The European option pricing problem with transaction costs is investigated for a risky asset price model with Levy jump By theaid of arbitrage pricing theory and the generalized Ito formula (which includes Poisson jump) the explicit solution to the risk assetprice model is given According to arbitrage-free principle we first discretize the continuous-time model Then in each small timeinterval the transaction costs are introduced By using theΔ-hedging strategy the explicit solutions of the European options pricingformula with transaction costs are given for the risky asset price model with Levy jump

1 Introduction

Recently the stochastic differential equation theory has foundmore andmore applications inmany fields such as finance [1ndash12] and control and filtering [13ndash25] The option pricing withtransaction costs has been one of the important problemsand received increasing research attention The developmentof the option pricing problem is reviewed as follows In[1] the hedge strategy has been introduced initially tosolve the option pricing problem with transaction costs In[5] the option pricing problem with transaction costs hasbeen transformed into a stochastic optimal control issueSubsequently a perfect hedge strategy has been proposed in[8] to deal with CRR model with transaction costs and adiscrete algorithm has been given In [26] the transactioncosts of European options have been proposed on the discretetime points for three scenarios that is without transactioncosts with proportional transaction costs and with concavetransaction costs By using the stochastic dominance theoryin [7] the upperlower limit has been given for Europeancall option with transaction costs as well as its correspondingvolatility In order to avoid solving a complicated stochasticoptimization problem in [27] an effective algorithm on

Markov chain has been proposed to obtain the Europeanoption price For the case of random volatility a nonlineardifferential equation has been constructed in [28] to price theEuropean option with transaction costs Some efforts havealso been made on the American option pricing problemsFor example the American option pricing problem has beenstudied for the jumpdiffusionmodel with transaction costs in[29] where the problem addressed has first been transformedinto a binary stochastic control problem and then solvednumerically

It should be pointed out that in the frictional financialmarket the models mentioned above have not taken therisks into account For the pricing problem with transactioncosts we also need to consider the risk asset pricing modelSince the B-S [2] option pricing formula put forward optionpricing has been an important part of financial mathematicsAs early as 1976 Merton [11] noticed that when somemajor message occurred the risky asset pricersquos changing wasdiscontinuous and pointed out that the risky asset was drivenby a Brownian motion and a jump diffusion model Aase[30] presented Ito process and random point process mixturemodel Scott [4] built a jump-diffusionmodel with stochasticvolatility and interest rate and gave the European option

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 513496 6 pageshttpdxdoiorg1011552014513496

2 Abstract and Applied Analysis

pricing formula Chanrsquos [31] cameupwith Levy processmodeland obtained the option pricing formula

In this paper the risk asset pricingmodel with Levy jumpdiffusion is considered and following Lelandrsquos [1] idea amore realistic pricing formula is given that hasmore practicalpotential

2 Problem Formulation and Preliminaries

The B-S model has been proposed in [2] where the B-Smodel based European option pricing formula has beenobtained In [11] the Poisson jump diffusion process hasbeen introduced to represent the stock price process andthe corresponding European option pricing formula has beenderived In this paper wewill investigate the European optionpricing problem for the Mertonrsquos model

The following assumption is needed

Assumption 1 Suppose that the following conditions aresatisfied

(i) risk-free rate 119903 is a constant(ii) there is no dividend(iii) there are no transaction costs(iv) there is no arbitrage opportunity

Consider the following asset model [32]

119889119878 (119905)

119878 (119905minus)= 120583119889119905 + 120590119889119882 (119905) + int

infin

minus1

119910119873 (119889119905 119889119910) (1)

where 119878(119905) is the price of risky asset suppose 119878(119905) has a jumpat time 119905 and then 119910 is the size of this jump which is also119910 = 119878(119905)minus119878(119905minus) 120583 represents the rate of return 120590 denotes thevolatility 119882(119905) is a standard Brownian motion 119873(119889119905 119889119910) isa Poisson measure with its intensity V(119889119905 119889119910)

From (1) we further have

119878 (119905) = 119878 (0) exp 120590119882 (119905) + (119903 minus1

21205902

) 119905

119873(119905)

prod

119894=1

(119880119894+ 1)

(2)

where 119903 is risk-free rateWe introduce the following lemmas that will be used to

obtain the main results

Lemma 2 (see [33]) Let 119884(119905) isin R be a Levy-Ito integrationdescribed as follows

119889119884 (119905) = 119866 (119905) 119889119905 + 119865 (119905) 119889119882 (119905) + int|119909|lt1

119867(119905 119909) (119889119905 119889119909)

+ int|119909|ge1

119870 (119905 119909)119873 (119889119905 119889119909)

(3)

where (119889119905 119889119909) is a compensated Poisson measure and thecontinuous part 119884119888(119905) of process satisfies

119889119884119888

(119905) = 119866 (119905) 119889119905 + 119865 (119905) 119889119882 (119905) (4)

Then for any 119891 isin 1198622

(R timesR) 119905 ge 0 119891(119884(119905)) is also a Levy-Itointegration and satisfies

119891 (119884 (119905)) minus 119891 (119884 (0))

= int

119905

0

120597119891

120597119910(119884 (119904minus)) 119889119884

119888

(119904)

+1

2int

119905

0

1205972

119891

1205971199102(119884 (119904minus)) 119889 [119884

119888

119884119888

] (119904)

+ int

119905

0

int|119909|ge1

[119891 (119884 (119904minus) + 119870 (119904 119909)) minus 119891 (119884 (119904minus))]119873 (119889119904 119889119909)

+ int

119905

0

int|119909|ge1

[119891 (119884 (119904minus) + 119867 (119904 119909)) minus 119891 (119884 (119904minus))] (119889119904 119889119909)

+ int

119905

0

int|119909|lt1

[119891 (119884 (119904minus) + 119867 (119904 119909)) minus 119891 (119884 (119904minus))

minus119867 (119904 119909)120597119891

120597119910(119884 (119904minus))] V (119889119909) 119889119904

(5)

Set

int

infin

0

int

infin

minus1

119910119873 (119889119905 119889119910) =

119873(119905)

sum

119895=1

119880119895 (6)

with probability and denote

119873(119905) =

119872

sum

119898=1

119873119898(119905) (7)

where119873(119905)1198731(119905) 119873

119872(119905) 0 le 119905 le 119879 are Poisson processes

with respective intensities 120582 1205821 120582

119872and (119880

119895)119895ge1

representsthe independent and identically distributed random variablestaking values in the set 119910

1 119910

119872

Defining 119901(119910119898) = 120582119898120582 we have the following lemma

Lemma 3 (see [34]) Let 119881(119905 119878) be the risk-neutral price of aEuropean call paying 119881(119879 119878(119879)) = (119878(119879) minus119870)

+

= max(119878(119879) minus119870 0) at time 119879 where119870 is a strike price If the risky asset pricemodel satisfies (1) then we have

120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(8)

Lemma 4 One has forall0 le 119905 lt 119879 the call optionrsquos final valuecondition is 119881(119879 119878(119879)) = (119878(119879) minus 119870)

+

= max(119878(119879) minus 119870 0)Then with the help of Lemma 234 in [34] one has

119881 (119905 119878 (119905)) = 119888 (119905 119878 (119905)) (9)

Abstract and Applied Analysis 3

where 119888(119905 119878(119905)) is the call option price given by

119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(10)

with 119889plusmn= (log(119878119870) + (119903 plusmn (12)120590

2

)(119879 minus 119905))120590radic119879 minus 119905

In an unstable financial market investors use somefinancial instruments to hedge risks and price its value Thebasic idea is to construct a portfolio to hedge risks SimilarlyΔ-hedge can also be used in the process of pricing optionWe take the risky asset price model following the geometricBrownian motion for example and describe the Δ-hedgeidea about the derivation of option pricing formula

Assume that the asset price model is given by119889119878 (119905)

119878 (119905)= 120583119889119905 + 120590119889119882 (119905) (11)

Suppose that there is no arbitrage opportunity We con-struct a portfolio Π = 119881 minus Δ119878 where Δ is the shares of riskyasset Note that in the interval (119905 119905 + 119889119905) Π is not risky andthe Δ is not changeable Hence the return rate of portfoliocan be given by

Π119905+119889119905

minus Π119905

Π119905

= 119903119889119905 (12)

Then we have119889119881 (119905) minus Δ119889119878 (119905) = 119903Π

119905119889119905 = 119903 (119881 minus Δ119878) 119889119905 (13)

On the other hand by using Ito formula it can beobtained that

119889119881 (119905) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878)119889119905 + 120590119878

120597119881

120597119878119889119882 (119905)

(14)Substituting (14) into (13) yields

(120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878minus Δ120583119878)119889119905

+ (120590119878120597119881

120597119878minus Δ120590119878)119889119882(119905) = 119903 (119881 minus Δ119878) 119889119905

(15)

Since there is no risk the coefficient of the random term119889119882(119905) in (15) should be zero that is

Δ =120597119881

120597119878 (16)

and hence (15) becomes120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 119903119878

120597119881

120597119878minus 119903119881 = 0 (17)

Finally according to the condition of the final value119881(119879 119878(119879)) = (119878(119879) minus 119870)

+

= max(119878(119879) minus 119870 0) we can obtainthe call option pricing formula

119888 (119905 119878) = 119878119873 (119889+) minus 119870119890

minus119903(119879minus119905)

119873(119889minus) (18)

where 119889plusmn= (log(119878119870) + (119903 plusmn (12)120590

2


3 Main Results

As an earlier derivation of option pricing is implementedunder some ldquoidealisedrdquo conditions its applications are lim-ited In order to loosen these restrictions the transactioncosts are introduced in the derivation of option pricing Forexample the hedging strategy has been considered in [2]when transaction costs occur at discrete time points and theexplicit formula of the call option price with transaction costshas been put forward In [29] the European option pricingproblem with the transaction costs has been studied for thejump diffusion model In this section we aim to give theEuropean option pricing formula with transaction costs forthe Levy jump case

In the interval (119905 119905 + 120575119905) with a sufficiently small 120575 gt 0the profit is given by

120575Π = 120575119881 minus Δ120575119878 minus 119896 |120592| 119878 (19)

where 119896 is the transaction cost ratio and 120592 is the numberof risky asset changing Hence it is easily known that thetransaction costs are 119896|120592|119878

Our main results are given as follows

Theorem 5 For the risky asset price model (1)119881(119905 119878) satisfies

120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(20)

Proof By using Taylorrsquos formula it follows from (16) that

120592 = Δ (119878 + 120575119878 119905 + 120575119905) minus Δ (119878 119905)

=120597119881

120597119878(119878 + 120575119878 119905 + 120575119905) minus

120597119881

120597119878(119878 119905)

=120597119881

120597119878(119878 119905) + 120575119878

1205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905)

+ 119900 (120575119905) minus120597119881

120597119878(119878 119905)

= 1205751198781205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)


120592 =1205972

119881

1205971198782(120583119878120575119905 + 120590119878120575119882 (119905)

+int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910))

+ 1205751199051205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)

(21)

Then we have

120592 asymp 1205901198781205972

119881

1205971198782120575119882 (119905) +

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910) (22)

According to no arbitrage principle E(120575Π) = 119903Π120575119905 andLemma 2 we have

E (120575Π) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782120575119882 (119905)

100381610038161003816100381610038161003816100381610038161003816

minus 119896119878E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

100381610038161003816100381610038161003816100381610038161003816

= (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E |120575119882 (119905)|

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816100381610038161003816

int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

10038161003816100381610038161003816100381610038161003816

(23)

Note that 120575119882(119905) sim 119873(0 120575119905) which implies

E |120575119882 (119905)| =1

radic2120587119878119905int

+infin

minusinfin

|119909| 119890minus11990922120575119905

119889119909

=2

radic2120587119878119905int

+infin

0

|119909| 119890minus11990922120575119905

119889119909

= radic2120575119905

120587

(24)

Therefore we have the following equation

120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(25)

This completes the proof

FromTheorem 5 the following theorem is obtained

Theorem 6 For 0 le 119905 lt 119879 the call option price withtransaction costs 119888(119905 119878(119905)) satisfies

119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(26)

where

119909 = 119878 (119905)

119895

prod

119894=1

(119880119894+ 1)

120591 = 119879 minus 119905

2

= 1205902

minus 2119896120590radic2

120587120575119905minus

2119896E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

119878120575119905

119889plusmn(120591 119909) =

log (119909119870) + (119903 minus (12) 2

) 120591

radic120591

(27)

Proof According to the definition of and Theorem 5 wehave

120597119881

120597119905+ 119903119878

120597119881

120597119878+1

22

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(28)

Then it follows from Lemmas 3 and 4 that

119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(29)

which completes the proof


Remark 7 In [9] the standard geometric Brownian motionmodel has been considered and the European option pricingformula with transaction costs has been obtained through theΔ-hedge The call option price given in [9] satisfies 119888(119905 119878) =

119878119873(1198891015840

1) minus 119870119890

minus119903(119879minus119905)

119873(1198891015840

2) where 1198891015840

1and 119889

1015840

2are the same as 119889

+

and 119889minusdefined in Theorem 6 but 1205901015840 is different from in

this paper In terms of comparison of the call option priceformula given in this paper and [9] it can be seen that theresults derived in this paper extend the ones in [9] and ourresults are more practically useful

4 Conclusions

In this paper the European option pricing problem withtransaction costs has been studied In order to make thepricing more practical we have chosen the Levy jumpdiffusion model instead of the standard geometric Brownianmotion model By using the Δ-hedged strategy the explicitcall option pricing formula has been obtained for the Levyjump case Our results have extended the ones in the existingliterature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National Natu-ral Science Foundation of China under Grant 60974030the Shanghai Rising-Star Program of China under Grant13QA1400100 and the Fundamental Research Funds for theCentral Universities of China

References

[1] E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 no 5 pp 1283ndash1301 1985

[2] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquo The Journal of Political Economy vol 81 no 3 pp637ndash654 1973

[3] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1976

[4] L O Scott ldquoPricing stock options in a jump-diffusion modelwith stochastic volatility and interest rates applications offourier inversion methodsrdquoMathematical Finance vol 7 no 4pp 413ndash426 1997

[5] M Davis and A Norman ldquoPortfolio selection with transactioncostsrdquo Mathematics of Operations Research vol 15 no 4 pp676ndash713 1990

[6] R Elliott and C Osakwe ldquoOption pricing for pure jumpprocesses with Markov switching compensatorsrdquo Finance andStochastics vol 10 no 2 pp 250ndash275 2006

[7] M George and S Perrakis ldquoStochastic dominance bounds onderivatives prices in a multiperiod economy with proportionaltransaction costsrdquo Journal of Economic Dynamics and Controlvol 26 no 7-8 pp 1323ndash1352 2002

[8] P P Boyle ldquoOption replication indiscrete time with transactioncostsrdquoThe Journal of Finance vol 47 pp 271ndash293 1992

[9] Q Li Option pricing with transaction costs [MS thesis]HuazhongUniversity of Science and Technology Hubei China2007

[10] R C Merton ldquoTheory of rational option pricingrdquo The BellJournal of Economics and Management Science vol 4 no 1 pp141ndash183 1973

[11] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[12] W Bailey and R M Stulz ldquoThe pricing of stock index optionsin a general equilibrium modelrdquo The Journal of Financial andQuantitative Analysis vol 24 no 1 pp 1ndash12 1989

[13] G L Wei L C Wang and F Han ldquoA gain-scheduled approachto fault-tolerant control for discrete-time stochastic delayedsystems with randomly occurring actuator faultsrdquo SystemsScience amp Control Engineering vol 1 no 1 pp 82ndash90 2013

[14] H Ahmada andTNamerikawa ldquoExtendedKalman filter-basedmobile robot localization with intermittent measurementsrdquoSystems Science amp Control Engineering vol 1 no 1 pp 113ndash1262013

[15] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013

[16] K El-Tawil and A Abou Jaoude ldquoStochastic and nonlinear-based prognostic modelrdquo Systems Science amp Control Engineer-ing vol 1 no 1 pp 66ndash81 2013

[17] M Darouach and M Chadli ldquoAdmissibility and control ofswitched discrete-time singular systemsrdquo Systems Science ampControl Engineering vol 1 no 1 pp 43ndash51 2013

[18] P Balasubramaniam and T Senthilkumar ldquoDelay-dependentrobust stabilization and 119867

infincontrol for uncertain stochastic

T-S fuzzy systems with discrete interval and distributed time-varying delaysrdquo International Journal of Automation and Com-puting vol 10 no 1 pp 18ndash31 2013

[19] P Zhou Y Wang Q Wang J Chen and D Duan ldquoStabilityand passivity analysis for Lurrsquoe singular systemswithMarkovianswitchingrdquo International Journal of Automation and Computingvol 10 no 1 pp 79ndash84 2013

[20] B Shen Z Wang and Y S Hung ldquoDistributed119867infin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[21] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[22] B Tang Q Zeng D He and Y Zhang ldquoRandom stabilizationof sampled-data control systems with nonuniform samplingrdquoInternational Journal of Automation and Computing vol 9 no5 pp 492ndash500 2012

[23] B Shen Z Wang and X Liu ldquoA stochastic sampled-dataapproach to distributed119867

infinFiltering in sensor networksrdquo IEEE

Transactions on Circuits and Systems I vol 58 no 9 pp 2237ndash2246 2011

[24] X Liang M Hou and G Duan ldquoOutput feedback stabiliza-tion of switched stochastic nonlinear systems under arbitraryswitchingsrdquo International Journal of Automation and Comput-ing vol 10 no 6 pp 571ndash577 2013


[25] Z Wang B Shen and X Liu ldquo119867infin

filtering with randomlyoccurring sensor saturations andmissingmeasurementsrdquoAuto-matica vol 48 no 3 pp 556ndash562 2012

[26] L Stettner ldquoOption pricing in theCRRmodelwith proportionaltransaction costs a cone transformation approachrdquo Applica-tiones Mathematicae vol 24 no 4 pp 475ndash514 1997

[27] M Monoyios ldquoOption pricing with transaction costs using aMarkov chain approximationrdquo Journal of Economic Dynamicsand Control vol 28 no 5 pp 889ndash913 2004

[28] P Amster C G Averbuj and M C Mariani ldquoStationarysolutions for two nonlinear BlackndashScholes type equationsrdquoApplied Numerical Mathematics vol 47 no 3-4 pp 275ndash2802003

[29] Q Wu ldquoEuropean options with transaction costsrdquo Journal ofTongji University vol 37 no 6 2008

[30] K K Aase ldquoContingent claims valuation when the securityprice is a combination of an Ito process and a random pointprocessrdquo Stochastic Processes and their Applications vol 28 no2 pp 185ndash220 1988

[31] T Chan ldquoPricing contingent claims on stocks diven by Levyprocessesrdquo The Annals of Applied Probability vol 9 no 2 pp504ndash528 1999

[32] H Yan Pricing option and EIAs when discrete dividends follow aMarkov-modulated jump diffusionmodel [MS thesis] DonghuaUniversity Shanghai China 2012

[33] R Cont and P Tankov Financial Modelling with Jump ProcessesChapman amp Hall Boca Raton Fla USA 2004

[34] S Shreve Stochastic Analysis for Finance Springer BerlinGermany 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


pricing formula Chanrsquos [31] cameupwith Levy processmodeland obtained the option pricing formula

In this paper the risk asset pricingmodel with Levy jumpdiffusion is considered and following Lelandrsquos [1] idea amore realistic pricing formula is given that hasmore practicalpotential

2 Problem Formulation and Preliminaries

The B-S model has been proposed in [2] where the B-Smodel based European option pricing formula has beenobtained In [11] the Poisson jump diffusion process hasbeen introduced to represent the stock price process andthe corresponding European option pricing formula has beenderived In this paper wewill investigate the European optionpricing problem for the Mertonrsquos model

The following assumption is needed

Assumption 1 Suppose that the following conditions aresatisfied

(i) risk-free rate 119903 is a constant(ii) there is no dividend(iii) there are no transaction costs(iv) there is no arbitrage opportunity

Consider the following asset model [32]

119889119878 (119905)

119878 (119905minus)= 120583119889119905 + 120590119889119882 (119905) + int

infin

minus1

119910119873 (119889119905 119889119910) (1)

where 119878(119905) is the price of risky asset suppose 119878(119905) has a jumpat time 119905 and then 119910 is the size of this jump which is also119910 = 119878(119905)minus119878(119905minus) 120583 represents the rate of return 120590 denotes thevolatility 119882(119905) is a standard Brownian motion 119873(119889119905 119889119910) isa Poisson measure with its intensity V(119889119905 119889119910)

From (1) we further have

119878 (119905) = 119878 (0) exp 120590119882 (119905) + (119903 minus1

21205902

) 119905

119873(119905)

prod

119894=1

(119880119894+ 1)

(2)

where 119903 is risk-free rateWe introduce the following lemmas that will be used to

obtain the main results

Lemma 2 (see [33]) Let 119884(119905) isin R be a Levy-Ito integrationdescribed as follows

119889119884 (119905) = 119866 (119905) 119889119905 + 119865 (119905) 119889119882 (119905) + int|119909|lt1

119867(119905 119909) (119889119905 119889119909)

+ int|119909|ge1

119870 (119905 119909)119873 (119889119905 119889119909)

(3)

where (119889119905 119889119909) is a compensated Poisson measure and thecontinuous part 119884119888(119905) of process satisfies

119889119884119888

(119905) = 119866 (119905) 119889119905 + 119865 (119905) 119889119882 (119905) (4)

Then for any 119891 isin 1198622

(R timesR) 119905 ge 0 119891(119884(119905)) is also a Levy-Itointegration and satisfies

119891 (119884 (119905)) minus 119891 (119884 (0))

= int

119905

0

120597119891

120597119910(119884 (119904minus)) 119889119884

119888

(119904)

+1

2int

119905

0

1205972

119891

1205971199102(119884 (119904minus)) 119889 [119884

119888

119884119888

] (119904)

+ int

119905

0

int|119909|ge1

[119891 (119884 (119904minus) + 119870 (119904 119909)) minus 119891 (119884 (119904minus))]119873 (119889119904 119889119909)

+ int

119905

0

int|119909|ge1

[119891 (119884 (119904minus) + 119867 (119904 119909)) minus 119891 (119884 (119904minus))] (119889119904 119889119909)

+ int

119905

0

int|119909|lt1

[119891 (119884 (119904minus) + 119867 (119904 119909)) minus 119891 (119884 (119904minus))

minus119867 (119904 119909)120597119891

120597119910(119884 (119904minus))] V (119889119909) 119889119904

(5)

Set

int

infin

0

int

infin

minus1

119910119873 (119889119905 119889119910) =

119873(119905)

sum

119895=1

119880119895 (6)

with probability and denote

119873(119905) =

119872

sum

119898=1

119873119898(119905) (7)

where119873(119905)1198731(119905) 119873

119872(119905) 0 le 119905 le 119879 are Poisson processes

with respective intensities 120582 1205821 120582

119872and (119880

119895)119895ge1

representsthe independent and identically distributed random variablestaking values in the set 119910

1 119910

119872

Defining 119901(119910119898) = 120582119898120582 we have the following lemma

Lemma 3 (see [34]) Let 119881(119905 119878) be the risk-neutral price of aEuropean call paying 119881(119879 119878(119879)) = (119878(119879) minus119870)

+

= max(119878(119879) minus119870 0) at time 119879 where119870 is a strike price If the risky asset pricemodel satisfies (1) then we have

120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(8)

Lemma 4 One has forall0 le 119905 lt 119879 the call optionrsquos final valuecondition is 119881(119879 119878(119879)) = (119878(119879) minus 119870)

+

= max(119878(119879) minus 119870 0)Then with the help of Lemma 234 in [34] one has

119881 (119905 119878 (119905)) = 119888 (119905 119878 (119905)) (9)



119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(10)


2




119878 (119905)= 120583119889119905 + 120590119889119882 (119905) (11)


Π119905+119889119905

minus Π119905

Π119905

= 119903119889119905 (12)


119905119889119905 = 119903 (119881 minus Δ119878) 119889119905 (13)


119889119881 (119905) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878)119889119905 + 120590119878

120597119881

120597119878119889119882 (119905)


(120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878minus Δ120583119878)119889119905

+ (120590119878120597119881

120597119878minus Δ120590119878)119889119882(119905) = 119903 (119881 minus Δ119878) 119889119905

(15)


Δ =120597119881

120597119878 (16)


120597119905+1

21205902

11987821205972

119881

1205971198782+ 119903119878

120597119881

120597119878minus 119903119881 = 0 (17)


+


119888 (119905 119878) = 119878119873 (119889+) minus 119870119890

minus119903(119879minus119905)

119873(119889minus) (18)


2


3 Main Results



120575Π = 120575119881 minus Δ120575119878 minus 119896 |120592| 119878 (19)




120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(20)


120592 = Δ (119878 + 120575119878 119905 + 120575119905) minus Δ (119878 119905)

=120597119881

120597119878(119878 + 120575119878 119905 + 120575119905) minus

120597119881

120597119878(119878 119905)

=120597119881

120597119878(119878 119905) + 120575119878

1205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905)

+ 119900 (120575119905) minus120597119881

120597119878(119878 119905)

= 1205751198781205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)


120592 =1205972

119881

1205971198782(120583119878120575119905 + 120590119878120575119882 (119905)

+int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910))

+ 1205751199051205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)

(21)

Then we have

120592 asymp 1205901198781205972

119881

1205971198782120575119882 (119905) +

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910) (22)


E (120575Π) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782120575119882 (119905)

100381610038161003816100381610038161003816100381610038161003816

minus 119896119878E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

100381610038161003816100381610038161003816100381610038161003816

= (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E |120575119882 (119905)|

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816100381610038161003816

int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

10038161003816100381610038161003816100381610038161003816

(23)


E |120575119882 (119905)| =1

radic2120587119878119905int

+infin

minusinfin

|119909| 119890minus11990922120575119905

119889119909

=2

radic2120587119878119905int

+infin

0

|119909| 119890minus11990922120575119905

119889119909

= radic2120575119905

120587

(24)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(25)




119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(26)

where

119909 = 119878 (119905)

119895

prod

119894=1

(119880119894+ 1)

120591 = 119879 minus 119905

2

= 1205902


120587120575119905minus

2119896E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

119878120575119905

119889plusmn(120591 119909) =

log (119909119870) + (119903 minus (12) 2

) 120591

radic120591

(27)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

22

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(28)


119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(29)




119878119873(1198891015840

1) minus 119870119890

minus119903(119879minus119905)

119873(1198891015840

2) where 1198891015840

1and 119889

1015840


+



4 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(10)


2




119878 (119905)= 120583119889119905 + 120590119889119882 (119905) (11)


Π119905+119889119905

minus Π119905

Π119905

= 119903119889119905 (12)


119905119889119905 = 119903 (119881 minus Δ119878) 119889119905 (13)


119889119881 (119905) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878)119889119905 + 120590119878

120597119881

120597119878119889119882 (119905)


(120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782+ 120583119878

120597119881

120597119878minus Δ120583119878)119889119905

+ (120590119878120597119881

120597119878minus Δ120590119878)119889119882(119905) = 119903 (119881 minus Δ119878) 119889119905

(15)


Δ =120597119881

120597119878 (16)


120597119905+1

21205902

11987821205972

119881

1205971198782+ 119903119878

120597119881

120597119878minus 119903119881 = 0 (17)


+


119888 (119905 119878) = 119878119873 (119889+) minus 119870119890

minus119903(119879minus119905)

119873(119889minus) (18)


2


3 Main Results



120575Π = 120575119881 minus Δ120575119878 minus 119896 |120592| 119878 (19)




120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(20)


120592 = Δ (119878 + 120575119878 119905 + 120575119905) minus Δ (119878 119905)

=120597119881

120597119878(119878 + 120575119878 119905 + 120575119905) minus

120597119881

120597119878(119878 119905)

=120597119881

120597119878(119878 119905) + 120575119878

1205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905)

+ 119900 (120575119905) minus120597119881

120597119878(119878 119905)

= 1205751198781205972

119881

1205971198782(119878 119905) + 120575119905

1205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)


120592 =1205972

119881

1205971198782(120583119878120575119905 + 120590119878120575119882 (119905)

+int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910))

+ 1205751199051205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)

(21)

Then we have

120592 asymp 1205901198781205972

119881

1205971198782120575119882 (119905) +

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910) (22)


E (120575Π) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782120575119882 (119905)

100381610038161003816100381610038161003816100381610038161003816

minus 119896119878E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

100381610038161003816100381610038161003816100381610038161003816

= (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E |120575119882 (119905)|

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816100381610038161003816

int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

10038161003816100381610038161003816100381610038161003816

(23)


E |120575119882 (119905)| =1

radic2120587119878119905int

+infin

minusinfin

|119909| 119890minus11990922120575119905

119889119909

=2

radic2120587119878119905int

+infin

0

|119909| 119890minus11990922120575119905

119889119909

= radic2120575119905

120587

(24)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(25)




119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(26)

where

119909 = 119878 (119905)

119895

prod

119894=1

(119880119894+ 1)

120591 = 119879 minus 119905

2

= 1205902


120587120575119905minus

2119896E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

119878120575119905

119889plusmn(120591 119909) =

log (119909119870) + (119903 minus (12) 2

) 120591

radic120591

(27)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

22

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(28)


119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(29)




119878119873(1198891015840

1) minus 119870119890

minus119903(119879minus119905)

119873(1198891015840

2) where 1198891015840

1and 119889

1015840


+



4 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120592 =1205972

119881

1205971198782(120583119878120575119905 + 120590119878120575119882 (119905)

+int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910))

+ 1205751199051205972

119881

120597119878120597119905(119878 119905) + 119900 (120575119905)

(21)

Then we have

120592 asymp 1205901198781205972

119881

1205971198782120575119882 (119905) +

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910) (22)


E (120575Π) = (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782120575119882 (119905)

100381610038161003816100381610038161003816100381610038161003816

minus 119896119878E

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

100381610038161003816100381610038161003816100381610038161003816

= (120597119881

120597119905+1

21205902

11987821205972

119881

1205971198782

+120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)])120575119905

minus 1198961205901198782

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E |120575119882 (119905)|

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816100381610038161003816

int

infin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)

10038161003816100381610038161003816100381610038161003816

(23)


E |120575119882 (119905)| =1

radic2120587119878119905int

+infin

minusinfin

|119909| 119890minus11990922120575119905

119889119909

=2

radic2120587119878119905int

+infin

0

|119909| 119890minus11990922120575119905

119889119909

= radic2120575119905

120587

(24)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

21205902

11987821205972

119881

1205971198782minus 119896120590119878

2

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

radic2

120587120575119905

minus 119896119878

100381610038161003816100381610038161003816100381610038161003816

1205972

119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816

E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

120575119905

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(25)




119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(26)

where

119909 = 119878 (119905)

119895

prod

119894=1

(119880119894+ 1)

120591 = 119879 minus 119905

2

= 1205902


120587120575119905minus

2119896E10038161003816100381610038161003816intinfin

minus1

119910119878 (119905minus)119873 (120575119905 120575119910)10038161003816100381610038161003816

119878120575119905

119889plusmn(120591 119909) =

log (119909119870) + (119903 minus (12) 2

) 120591

radic120591

(27)


120597119881

120597119905+ 119903119878

120597119881

120597119878+1

22

11987821205972

119881

1205971198782

+ 120582[

119872

sum

119898=1

119901 (119910119898) 119881 (119905 (119910

119898+ 1) 119878) minus 119881 (119905 119878)] minus 119903119881 = 0

(28)


119888 (119905 119878 (119905)) =

infin

sum

119895=0

119890minus120582(119879minus119905)

120582119895

(119879 minus 119905)119895

119895E [119909119873 (119889

+(120591 119909))

minus119870119890minus119903120591

119873(119889minus(120591 119909))]

(29)




119878119873(1198891015840

1) minus 119870119890

minus119903(119879minus119905)

119873(1198891015840

2) where 1198891015840

1and 119889

1015840


+



4 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878119873(1198891015840

1) minus 119870119890

minus119903(119879minus119905)

119873(1198891015840

2) where 1198891015840

1and 119889

1015840


+



4 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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